Most electronic circuits may require a stable direct current (DC) voltage reference, particularly with regard to fluctuations of working temperature for the circuits. Usually, such stable voltage reference circuits are bandgap voltage generators that are based upon the property of a bipolar transistor to produce a base-emitter voltage with well known temperature dependence.
According to a theoretical analysis in the article: Yannys P. Tsividis, “Accurate analysis of temperature effects in IC-VBE characteristics with application to bandgap reference sources”, IEEE Journal of solid-state circuits, Vol. SC-15, No. 6, December 1980, pages 1076-1084, the following equation holds:
                                          V            BE                    ⁡                      (            T            )                          =                              V                          BG              ⁢                                                          ⁢              0                                -                                    (                                                V                                      BG                    ⁢                                                                                  ⁢                    0                                                  -                                                      V                    BE                                    ⁡                                      (                                          T                      REF                                        )                                                              )                        ·                          T                              T                REF                                              -                      α            ·                          V              T                        ·                          ln              ⁡                              (                                  T                  /                                      T                    REF                                                  )                                                                        (        1        )            
where VBE is the base-emitter voltage, VBG0 is the bandgap voltage expected at a null temperature, TREF is a reference temperature, α is a coefficient, and VT is the voltage equivalent of temperature. The following equation holds for VT:
      V    T    =      kT    q  
Neglecting the (generally) small term α·VT·ln(T/TREF), VBE voltage is complementary to the absolute temperature (CTAT). In literature, two main classes of bandgap generators are disclosed:                1. First Order Bandgap: this is the oldest type of voltage reference. Its architecture used since 1960's. It typically has a temperature coefficient TC=50 ppm/° C. and an absolute value spread of 12 mV, in a 190° C. temperature range.        2. Second Order Bandgap: this type has been used for the last 10-15 years. It has a typical temperature coefficient TC=15 ppm/° C. and absolute value spread of 3 mV, in a 190° C. temperature range.        
In known bandgap voltage reference generators, for example, the generator disclosed in U.S. Pat. No. 4,249,122 to Widlar, a pair of transistors are operated at different current densities and are coupled to generate a voltage that is proportional to the difference between the base-emitter voltages of the two transistors. This difference voltage has a positive temperature coefficient, i.e. the difference voltage is proportional to the absolute temperature (PTAT) of the circuit. The PTAT voltage provided by the difference in the base-emitter voltages is properly scaled and summed with the complementary to absolute temperature voltage of one of the transistors to generate a stable bandgap voltage reference.
In first-order bandgap compensation, the first derivative of the base-emitter voltage with respect to temperature is nullified in correspondence to a reference temperature TREF, as shown in FIG. 1, thus the generated bandgap voltage varies with the working absolute temperature T, assuming a typical peak value of 1.22V with a typical maximum fluctuation of about 12 mV. The term α·VT·ln(T/TREF) is the cause of the residual temperature dependency after a first-order compensation.
In widely diffused second order bandgap voltage generators, a voltage proportional to the square absolute temperature (PSTAT) is used to compensate the second order term of the Taylor expansion of a α·VT·ln(T/TREF), such to nullify at the reference temperature TREF the first derivative and the second derivative of the output voltage VOUT with respect to the absolute temperature, obtaining a voltage-temperature characteristic as shown by way of example in FIG. 2.
In other second order bandgap voltage generators, a nonlinear current is generated. This current is proportional to T*ln(T/Tref) and it is added to compensate for the term α·VT·ln(T/TREF). An exemplary architecture implementing such a second order bandgap compensation is shown in FIG. 3a and is disclosed in the article by Guang Ge, Cheng Zhang, Gian Hoogzaad, Kofi Makinwa, “A single-trim CMOS bandgap reference with a 3σ inaccuracy of ±0.15% from −40° C. to 125° C.,” 2010 IEEE International Solid-State Circuits Conference, session 4, analog techniques, 4.3, pages 78-80. The current is generated by the difference of two bipolar's Vbe: one of them is biased with a PTAT current, while the other transistor is biased with a current constant versus temperature. An exemplary variation of the output voltage with temperature for the circuit of FIG. 3a is shown in FIG. 3b. Typically, voltage fluctuations with temperature are relatively reduced. Other architectures that include a second order bandgap compensation are disclosed in U.S. Pat. Nos. 6,828,847, 7,598,799, 7,514,987, and 7,583,135.
Even if voltage fluctuations with temperature are limited in a smaller range than that of first-order bandgap voltage generators, these architectures may be complicated to realize and/or cannot accurately and independently adjust the PTAT and logarithmic terms. In other words, the generated bandgap voltage, after the trimming procedure, may vary greatly in temperature ranges from −40° C. up to 150° C.